Abstract:
We apply one of the formalisms of noncommutative geometry to ℝN q , the quantum space covariant under the quantum group SO q (N). Over ℝN q there are two SO q (N)-covariant differential calculi. For each we find a frame, a metric and two torsion-free covariant derivatives which are metric compatible up to a conformal factor and which have a vanishing linear curvature. This generalizes results found in a previous article for the case of ℝ3 q . As in the case N=3, one has to slightly enlarge the algebra ℝN q ; for N odd one needs only one new generator whereas for N even one needs two. As in the particular case N=3 there is a conformal ambiguity in the natural metrics on the differential calculi over ℝN q . While in our previous article the frame was found “by hand”, here we disclose the crucial role of the quantum group covariance and exploit it in the construction. As an intermediate step, we find a homomorphism from the cross product of ℝN q with U q so(N) into ℝN q , an interesting result in itself.
Similar content being viewed by others
Author information
Authors and Affiliations
Additional information
Received: 4 March 2000 / Accepted: 11 October 2000
Rights and permissions
About this article
Cite this article
Cerchiai, B., Fiore, G. & Madore, J. Geometrical Tools for Quantum Euclidean Spaces. Commun. Math. Phys. 217, 521–554 (2001). https://doi.org/10.1007/PL00005553
Issue Date:
DOI: https://doi.org/10.1007/PL00005553